Maissam Barkeshli , Xiao-Liang Qi : Synthetic Topological Qubits in Conventional Bilayer Quantum Hall Systems , Phys. Rev. X 4 (2014) 041035 [doi:10.1103/PhysRevX.4.041035 , arXiv:1302.2673 ]
Roger S. K. Mong et al.: Universal Topological Quantum Computation from a Superconductor-Abelian Quantum Hall Heterostructure , Phys. Rev. X 4 (2014) 011036 [doi:10.1103/PhysRevX.4.011036 , arXiv:1307.4403 ]
D. V. Averin, V. J. Goldman: Quantum computation with quasiparticles of the Fractional Quantum Hall Effect , Solid State Communications 121 1 (2001) 25-28 [doi:10.1016/S0038-1098(01)00447-1 , arXiv:cond-mat/0110193 ]
A. P. Balachandran, A. Ibort, G. Marmo, M. Martone: Quantum Geons and Noncommutative Spacetimes
0 → π 1 ( Maps ) → π 2 ( Maps ⫽ Diff ) → B ( ℤ × ℤ ) → π 1 ( Maps ) → π 1 ( Maps ⫽ Diff ) → B MCG → ℤ → ℤ → π 0 ( B Diff )
0
\to
\pi_1(Maps)
\to
\pi_2\big(Maps \sslash Diff\big)
\to
B (\mathbb{Z} \times \mathbb{Z})
\to
\pi_1(Maps)
\to
\pi_1\big(Maps \sslash Diff\big)
\to
B MCG
\to
\mathbb{Z}
\to
\mathbb{Z}
\to
\pi_0(B Diff)
Annular braid group
The annular braid group Br n ( An ) Br_n(An) on n ∈ ℕ n \in \mathbb{N} strands is the surface braid group of the annulus , hence the fundamental group of the [configuration space of points|configuration space of -points]] inside the annulus .
Properties
Proposition
The annular braid group is isomorphic to a semidirect product
Br n ( An ) ≃ Br n aff ⋊ ℤ
Br_n(An)
\,\simeq\,
Br_n^{aff} \rtimes \mathbb{Z}
of the the affine braid group? with the group of integers , the latter generated by the braid which exhibits a 1-step cyclic permutation .
References
Richard P. Kent, David Pfeifer: A Geometric and algebraic description a Of annular braid groups , International Journal of Algebra and Computation 12 01n02 (2002) 85-97 [doi:10.1142/S0218196702000997 ]
Paolo Bellingeri, Arnaud Bodin: The braid group of a necklace , Mathematische Zeitschrift 283 3 (2014) [arXiv:1404.2511 , doi:10.1007/s00209-016-1630-0 ]
Asymptotic gauge groups
Given a gauge theory (and/or gravity ) on a spacetime with asymptotic boundary , certain would-be gauge transformations (diffeomorphisms ) that act non-trivially on asymptotic “boundary data” may in fact be identified as physically observable global symmetries and hence have, in contrast to actual gauge symmetries , “direct empirical significance” (DES, Teh 2016 ).
A key example is (symmetry generated by) the ADM mass and generally the BMS group? of asymptotic symmetries in asymptotically flat spacetimes .
Up to technical fine-print (cf. Borsboom & Posthuma 2015 ) a group of asymptotic symmetries is the coset space of all gauge symmetries that respect boundary data by the subgroup of bulk gauge transformations which act as the identity map on the asymptotic boundary (cf. Strominger 2018 (2.10.1) , Borsboom & Posthuma 2015 p 2 ):
AsymptoticSymmetries = BoundaryAdmissibleGaugeSymmetries BulkGaugeSymmetriesTrivialAtBoundary
AsymptoticSymmetries
\;=\;
\frac{
BoundaryAdmissibleGaugeSymmetries
}{
BulkGaugeSymmetriesTrivialAtBoundary
}
Hence if the bulk gauge symmetries form a normal subgroup then the asymptotic symmetries form a quotient group characterized by a short exact sequence of the form
1 → BulkGaugeSymmetriesTrivialAtBoundary ⟶ BoundaryAdmissibleGaugeSymmetries ⟶ AsymptoticSymmetries → 1 .
1 \to
BulkGaugeSymmetriesTrivialAtBoundary
\longrightarrow
BoundaryAdmissibleGaugeSymmetries
\longrightarrow
AsymptoticSymmetries
\to
1
\,.
References
pdf
J. Tolar: On Clifford groups in quantum computing [arXiv:1810.10259 ]
Definition
By the spherical braid group Br n ( S 2 ) Br_n(S^2) , for n ∈ ℕ n \in \mathbb{N} , one means the surface braid group
Br n ( S 2 ) ≃ π 1 Conf n ( S 2 ) ,
Br_n(S^2)
\;\simeq\;
\pi_1
Conf_n(S^2)
\,,
where the surface in question is the 2-sphere S 2 S^2 . Hence the surface braid group is the fundamental group π 1 ( − ) \pi_1(-) of the configuration space of
n
n
-points , Conf n ( − ) Conf_n(-) , on the 2-sphere .
Properties
Proposition
The spherical braid group is the quotient group of the ordinary braid group by one further relation:
Br n ( S 2 ) ≃ Br n / ( ( b 1 b 2 ⋯ b n − 1 ) ( b n − 1 ⋯ b 2 b 1 ) ,
Br_n(S^2)
\;\simeq\;
Br_n/
\big( (b_1 b_2 \cdots b_{n-1})(b_{n-1} \cdots b_2 b_1)
\,,
where the b i b_i denote the Artin braid generators .
Moreover, the canonical map from the plain braid group to the symmetric group factors through this quotient map to the spherical braid group
(
Fadell & Van Buskirk 1961 p 245, 255 , cf.
Tan 2024 §3.1 )
References
π 2 S 2 → π 1 Maps * ( S 2 , S 2 ) → π 1 Maps ( S 2 , S 2 ) → π 1 S 2 → π 0 Maps * ( S 2 , S 2 ) → π 0 Maps ( S 2 , S 2 ) → π 0 S 2
\pi_2 S^2
\to
\pi_1
\Maps^\ast(S^2, S^2)
\to
\pi_1
Maps(S^2, S^2)
\to
\pi_1 S^2
\to
\pi_0
\Maps^\ast(S^2, S^2)
\to
\pi_0
Maps(S^2, S^2)
\to
\pi_0 S^2
Proof
For definiteness of computation, when group averaging we will be cycling through the elements of Sym 3 Sym_3 in this order:
Sym 3 = { ( 1 2 3 ) , ( 1 3 2 ) , ( 2 1 3 ) , ( 2 3 1 ) , ( 3 1 2 ) , ( 3 2 1 ) } .
Sym_3
\;=\;
\left\{
\begin{array}{l}
(1 2 3)
,\,\\
(1 3 2)
,\,\\
(2 1 3)
,\,\\
(2 3 1)
,\,\\
(3 1 2)
,\,\\
(3 2 1)
\end{array}
\right\}
\,.
Let { e 1 , e 2 , e 3 } \big\{e_1, e_2, e_3\big\} denote the canonical linear basis of the defining representation, with group action given by
σ ⋅ e i ≔ e σ ( i ) .
\sigma \cdot e_i \;\coloneqq\; e_{\sigma(i)}
\,.
Then a linear basis for the standard representation inside the defining representation is:
[ 1 0 ] ≔ e 1 − e 3 [ 0 1 ] ≔ e 2 − e 3 .
\begin{array}{ccc}
\left[\begin{array}{c}1 \\ 0\end{array}\right]
\;\coloneqq\;
e_1 - e_3
\\
\left[\begin{array}{c}0 \\ 1\end{array}\right]
\;\coloneqq\;
e_2 - e_3
\mathrlap{\,.}
\end{array}
The group-averaged inner product of these basis elements is found to be:
⟨ [ 1 0 ] , [ 0 1 ] ⟩ = ( [ 1 0 ] ⋅ [ 0 1 ] + [ 1 − 1 ] ⋅ [ 0 − 1 ] + [ 0 1 ] ⋅ [ 1 0 ] + [ − 1 1 ] ⋅ [ − 1 0 ] + [ 0 − 1 ] ⋅ [ 1 − 1 ] + [ − 1 0 ] ⋅ [ − 1 1 ] ) / 6 = 4 / 6 = 2 / 3
\left\langle
\left[
\begin{array}{c}
1 \\ 0
\end{array}
\right]
,\,
\left[
\begin{array}{c}
0 \\ 1
\end{array}
\right]
\right\rangle
\;=\;
\left(
\;\;
\begin{array}{l}
\left[
\begin{array}{c}
1
\\
0
\end{array}
\right]
\cdot
\left[
\begin{array}{c}
0
\\
1
\end{array}
\right]
+
\left[
\begin{array}{c}
1
\\
-1
\end{array}
\right]
\cdot
\left[
\begin{array}{c}
0
\\
-1
\end{array}
\right]
+
\left[
\begin{array}{c}
0
\\
1
\end{array}
\right]
\cdot
\left[
\begin{array}{c}
1
\\
0
\end{array}
\right]
+
\left[
\begin{array}{c}
-1
\\
1
\end{array}
\right]
\cdot
\left[
\begin{array}{c}
-1
\\
0
\end{array}
\right]
+
\left[
\begin{array}{c}
0
\\
-1
\end{array}
\right]
\cdot
\left[
\begin{array}{c}
1
\\
-1
\end{array}
\right]
+
\left[
\begin{array}{c}
-1
\\
0
\end{array}
\right]
\cdot
\left[
\begin{array}{c}
-1
\\
1
\end{array}
\right]
\end{array}
\right)/6
\;=\;
4/6
\;=\;
2/3
⟨ [ 0 1 ] , [ 0 1 ] ⟩ = ( [ 0 1 ] ⋅ [ 0 1 ] + [ 0 − 1 ] ⋅ [ 0 − 1 ] + [ 1 0 ] ⋅ [ 1 0 ] + [ − 1 0 ] ⋅ [ − 1 0 ] + [ 1 − 1 ] ⋅ [ 1 − 1 ] + [ − 1 1 ] ⋅ [ − 1 1 ] ) / 6 = 8 / 6 = 4 / 3
\left\langle
\left[
\begin{array}{c}
0 \\ 1
\end{array}
\right]
,\,
\left[
\begin{array}{c}
0 \\ 1
\end{array}
\right]
\right\rangle
\;=\;
\left(
\;\;
\begin{array}{l}
\left[
\begin{array}{c}
0
\\
1
\end{array}
\right]
\cdot
\left[
\begin{array}{c}
0
\\
1
\end{array}
\right]
+
\left[
\begin{array}{c}
0
\\
-1
\end{array}
\right]
\cdot
\left[
\begin{array}{c}
0
\\
-1
\end{array}
\right]
+
\left[
\begin{array}{c}
1
\\
0
\end{array}
\right]
\cdot
\left[
\begin{array}{c}
1
\\
0
\end{array}
\right]
+
\left[
\begin{array}{c}
-1
\\
0
\end{array}
\right]
\cdot
\left[
\begin{array}{c}
-1
\\
0
\end{array}
\right]
+
\left[
\begin{array}{c}
1
\\
-1
\end{array}
\right]
\cdot
\left[
\begin{array}{c}
1
\\
-1
\end{array}
\right]
+
\left[
\begin{array}{c}
-1
\\
1
\end{array}
\right]
\cdot
\left[
\begin{array}{c}
-1
\\
1
\end{array}
\right]
\end{array}
\right)/6
\;=\;
8/6
\;=\;
4/3
⟨ [ 1 0 ] , [ 1 0 ] ⟩ = ( [ 1 0 ] ⋅ [ 1 0 ] + [ 1 − 1 ] ⋅ [ 1 − 1 ] + [ 0 1 ] ⋅ [ 0 1 ] + [ − 1 1 ] ⋅ [ − 1 1 ] + [ 0 − 1 ] ⋅ [ 0 − 1 ] + [ − 1 0 ] ⋅ [ − 1 0 ] ) / 6 = 8 / 6 = 4 / 3
\left\langle
\left[
\begin{array}{c}
1 \\ 0
\end{array}
\right]
,\,
\left[
\begin{array}{c}
1 \\ 0
\end{array}
\right]
\right\rangle
\;=\;
\left(
\;\;
\begin{array}{l}
\left[
\begin{array}{c}
1
\\
0
\end{array}
\right]
\cdot
\left[
\begin{array}{c}
1
\\
0
\end{array}
\right]
+
\left[
\begin{array}{c}
1
\\
-1
\end{array}
\right]
\cdot
\left[
\begin{array}{c}
1
\\
-1
\end{array}
\right]
+
\left[
\begin{array}{c}
0
\\
1
\end{array}
\right]
\cdot
\left[
\begin{array}{c}
0
\\
1
\end{array}
\right]
+
\left[
\begin{array}{c}
-1
\\
1
\end{array}
\right]
\cdot
\left[
\begin{array}{c}
-1
\\
1
\end{array}
\right]
+
\left[
\begin{array}{c}
0
\\
-1
\end{array}
\right]
\cdot
\left[
\begin{array}{c}
0
\\
-1
\end{array}
\right]
+
\left[
\begin{array}{c}
-1
\\
0
\end{array}
\right]
\cdot
\left[
\begin{array}{c}
-1
\\
0
\end{array}
\right]
\end{array}
\right)/6
\;=\;
8/6
\;=\;
4/3
From this an orthonormal basis for the averaged inner product is
| 0 ⟩ ≔ 1 2 [ 1 1 ] = 1 2 ( e 1 + e 2 ) − e 3 | 1 ⟩ ≔ 3 2 [ 1 − 1 ] = 3 2 ( e 1 − e 2 ) .
\begin{array}{l}
{\vert 0 \rangle}
\;\coloneqq\;
\tfrac{1}{2}
\left[
\begin{array}{c}
1
\\
1
\end{array}
\right]
\;=\;
\tfrac{1}{2}(e_1 + e_2) - e_3
\\
{\vert 1 \rangle}
\;\coloneqq\;
\tfrac{\sqrt{3}}{2}
\left[
\begin{array}{c}
1
\\
-1
\end{array}
\right]
\;=\;
\tfrac{\sqrt{3}}{2}(e_1 - e_2)
\,.
\end{array}
On this bases the action of ( 213 ) (213) is
ρ ( 213 ) ( | 0 ⟩ ) = | 0 ⟩
\rho(213)\big({\vert 0 \rangle}\big)
\;=\;
{\vert 0 \rangle}
ρ ( 213 ) ( | 1 ⟩ ) = − | 1 ⟩
\rho(213)\big({\vert 1 \rangle}\big)
\;=\;
-{\vert 1 \rangle}
and hence ( 213 ) (213) acts as the Pauli Z-gate :
ρ ( 213 ) = ( 1 0 0 − 1 ) = Z
\rho(213)
\;=\;
\left(
\begin{array}{cc}
1 & 0
\\
0 & -1
\end{array}
\right)
\;=\;
Z
On the other hand, the action of ( 132 ) (132) is found to be
ρ ( 132 ) ( | 0 ⟩ ) = ρ ( 132 ) ( 1 2 [ 1 1 ] ) = 1 2 [ 1 − 2 ] = − 1 2 | 0 ⟩ + 3 2 | 1 ⟩
\rho(132)\big({\vert 0 \rangle}\big)
\;=\;
\rho(132)
\left(
\tfrac{1}{2}
\left[
\begin{array}{c}
1
\\
1
\end{array}
\right]
\right)
\;=\;
\tfrac{1}{2}
\left[
\begin{array}{c}
1
\\
-2
\end{array}
\right]
\;=\;
-\tfrac{1}{2} {\vert 0 \rangle}
+ \tfrac{\sqrt{3}}{2} {\vert 1 \rangle}
ρ ( 132 ) ( | 1 ⟩ ) = ρ ( 132 ) ( 3 2 [ 1 − 1 ] ) = 3 2 [ 1 0 ] = 3 2 | 0 ⟩ + 1 2 | 1 ⟩
\rho(132)\big({\vert 1 \rangle}\big)
\;=\;
\rho(132)
\left(
\tfrac{\sqrt{3}}{2}
\left[
\begin{array}{c}
1
\\
-1
\end{array}
\right]
\right)
\;=\;
\tfrac{\sqrt{3}}{2}
\left[
\begin{array}{c}
1
\\
0
\end{array}
\right]
\;=\;
\tfrac{\sqrt{3}}{2} {\vert 0 \rangle}
+
\tfrac{1}{2} {\vert 1 \rangle}
and hence
ρ ( 132 ) = ( − 1 / 2 3 / 2 3 / 2 1 / 2 ) = ( − 1 0 0 1 ) ( 1 / 2 − 3 / 2 3 / 2 1 / 2 ) = ( − 1 0 0 1 ) ( cos ( π / 3 ) − sin ( π / 3 ) sin ( π / 3 ) cos ( π / 3 ) ) = − Z ∘ R y ( 2 π / 3 )
\rho(132)
\;=\;
\left(
\begin{array}{c}
-1/2 & \sqrt{3}/2
\\
\sqrt{3}/2 & 1/2
\end{array}
\right)
\;=\;
\left(
\begin{array}{c}
-1 & 0
\\
0 & 1
\end{array}
\right)
\left(
\begin{array}{c}
1/2 & -\sqrt{3}/2
\\
\sqrt{3}/2 & 1/2
\end{array}
\right)
\;=\;
\left(
\begin{array}{c}
-1 & 0
\\
0 & 1
\end{array}
\right)
\left(
\begin{array}{c}
cos(\pi/3) & -sin(\pi/3)
\\
sin(\pi/3) & cos(\pi/3)
\end{array}
\right)
\;=\;
-
Z \circ R_y(2\pi/3)
end
\lnebreak
The action of ( 231 ) (231)
( 231 ) | 0 ⟩ = 1 2 [ − 2 1 ] = − 1 2 | 0 ⟩ − 3 2 | 1 ⟩
(231) {\vert 0 \rangle}
\;=\;
\tfrac{1}{2}
\left[
\begin{array}{c}
-2 \\ 1
\end{array}
\right]
\;=\;
-\tfrac{1}{2} {\vert 0 \rangle}
-\tfrac{\sqrt{3}}{2} {\vert 1 \rangle}
( 231 ) | 1 ⟩ = 3 2 [ 0 1 ] = 3 2 | 0 ⟩ − 1 2 | 1 ⟩
(231) {\vert 1 \rangle}
\;=\;
\tfrac{\sqrt{3}}{2}
\left[
\begin{array}{c}
0 \\ 1
\end{array}
\right]
\;=\;
\tfrac{\sqrt{3}}{2} {\vert 0 \rangle}
- \tfrac{1}{2} {\vert 1 \rangle}
Hence
( 231 ) = [ − 1 / 2 3 / 2 − 3 / 2 − 1 / 2 ]
(231)
\;=\;
\left[
\begin{array}{cc}
-1/2 & \sqrt{3}/2
\\
-\sqrt{3}/2 & - 1/2
\end{array}
\right]
The action of ( 321 ) (321)
( 321 ) | 0 ⟩ = ( 321 ) 1 2 [ 1 1 ] = 1 2 [ − 2 1 ] = − 1 2 | 0 ⟩ − 3 2 | 1 ⟩
(321) {\vert 0 \rangle}
\;=\;
(321)
\tfrac{1}{2}
\left[
\begin{array}{c}
1 \\ 1
\end{array}
\right]
\;=\;
\tfrac{1}{2}
\left[
\begin{array}{c}
-2 \\ 1
\end{array}
\right]
\;=\;
-\tfrac{1}{2} {\vert 0 \rangle}
- \tfrac{\sqrt{3}}{2} {\vert 1 \rangle}
( 321 ) | 1 ⟩ = ( 321 ) 3 2 [ 1 − 1 ] = 3 2 [ 0 − 1 ] = 3 2 | 0 ⟩ − 1 2 | 1 ⟩
(321) {\vert 1 \rangle}
\;=\;
(321)
\tfrac{\sqrt{3}}{2}
\left[
\begin{array}{c}
1 \\ -1
\end{array}
\right]
\;=\;
\tfrac{\sqrt{3}}{2}
\left[
\begin{array}{c}
0 \\ -1
\end{array}
\right]
\;=\;
\tfrac{\sqrt{3}}{2} {\vert 0 \rangle}
- \tfrac{1}{2} {\vert 1 \rangle}
Hence ( 321 ) (321) acts as
( 321 ) = [ − 1 / 2 3 / 2 − 3 / 2 − 1 / 2 ] = − [ 1 / 2 − 3 / 2 3 / 2 1 / 2 ] = − R y ( 2 π / 3 )
(321)
\;=\;
\left[
\begin{array}{c}
-1/2 & \sqrt{3}/2
\\
-\sqrt{3}/2 & -1/2
\end{array}
\right]
\;=\;
-
\left[
\begin{array}{c}
1/2 & -\sqrt{3}/2
\\
\sqrt{3}/2 & 1/2
\end{array}
\right]
\;=\;
- R_y(2\pi/3)
Defect anyons
Controlling anyons
Loop space of the 2-sphere
On the loop space Ω S 2 \Omega S^2 of the 2-sphere
in relation to braid groups
Frederick R. Cohen , J. Wu: On Braid Groups, Free Groups, and the Loop Space of the 2-Sphere , in: Categorical Decomposition Techniques in Algebraic Topology , in Progress in Mathematics 215 , Birkhäuser (2003) 93-105 [doi:10.1007/978-3-0348-7863-0_6 ]
On Ω S 2 ≃ B Ω 2 S 2 \Omega S^2 \,\simeq\, B \Omega^2 S^2 regarded as a classifying space (for “l \mathbf{l} ine” bundles):